Method and system for selecting parameters of a seismic source array

ABSTRACT

A method for selecting parameters of a seismic source array comprising a plurality of source elements each having a notional source spectrum is described, the method comprising calculating a ghost response function of the array; calculating directivity effects of the array; and adjusting the parameters of the array such that the directivity effects of the array are compensated by the ghost response to minimize angular variation of a far field response in a predetermined frequency range. A method for determining a phase center of a seismic source array is also related, the method comprising calculating a far field spectrum of the array at predetermined spherical angles, and minimizing the phase difference between the farfield spectra within a predetermined frequency range by adjusting a vertical reference position from which the spherical angles are defined.

FIELD

The present invention relates to a method and system for selectingparameters of a seismic source array, more particularly, selectingphysical parameters that will minimize angular variation of the farfield spectrum. The invention also relates to a method and system fordetermining the phase centre of a seismic source array that minimizesthe angular phase variation of the array.

BACKGROUND

Seismic sources usually comprise arrays of individual seismic sourceelements. The most common marine seismic source elements are airguns,but other elements such as vibrators, waterguns and sparkers etc. mayalso be used. The seismic source elements behave individually as pointsources over the bandwidth of interest and are each characterized by anotional source signature, sometimes called the monopole sourcefunction.

Seismic source arrays exhibit directivity. This directivity may producedirectivity patterns that are determined by the notional sourcesignatures, the positions, and the activation times of the sourceelements in the array. The reflected signal from the sea surface maystrongly affect the directivity pattern

For some conditions, it is desirable that the source array should behaveas closely as possible to a monopole source. Today's commerciallyavailable sources try to achieve this by reducing the array size suchthat the maximum array dimension is considerable smaller than theshortest wavelength of interest. However, this will not result in amonopole source spectrum when the sea surface reflection is taken intoaccount. The resulting source will be a dipole. Embodiments of thepresent invention may provide a method for designing an improvedmonopole source configuration.

SUMMARY

The present invention provides a method for selecting parameters of aseismic source array comprising a plurality of source elements eachhaving a notional source spectrum, the method comprising:

-   -   calculating a ghost response function of the array;    -   calculating directivity effects of the array; and    -   adjusting the parameters of the array such that the directivity        effects of the array are compensated by the ghost response to        minimize angular variation of a far field response in a        predetermined frequency range.

Preferably, the depths of the source elements are selected such that afirst notch in the ghost response is above the predetermined frequencyband.

In one embodiment, the array comprises a single layer of source elementsat a common depth, and the parameters comprise the length to depth ratioof the source array. Preferably, the length to depth ratio is 1.5 to 3.

In another embodiment, the array comprises a plurality of layers, eachlayer comprising a plurality of source elements having substantially thesame depth. Preferably, the minimum wavelength of the predeterminedfrequency band is greater than 4/3 of the maximum source element depth,i.e. d_(max)/λ<0.75, more preferably the minimum wavelength of thepredetermined frequency band is greater than two times the maximumsource element depth, i.e. d_(max)/λ<0.5 Preferably, the depth ratio ofthe layers is adjusted such that a first notch in the ghost response isabove the predetermined frequency band. Preferably, the array comprisestwo layers of source elements and the depth ratio of the two layers isin the range 0.25 to 0.6, more preferably 0.3 to 0.5. Preferably, thenotional source spectra of the elements at each depth layer aresubstantially identical. Preferably, the length to depth ratio of eachlayer is less than 2.

In another embodiment, the element positions substantially form avertical line array.

In a preferred embodiment, the positions of the source elements haverotational symmetry in azimuth of order three or greater in thehorizontal plane.

Preferably, the angular variation is minimized within the range oftake-off angles of 0 to 40 degrees.

Preferably, the predetermined frequency range is 0 to 150 Hz.

Preferably, the method further comprises:

-   -   calculating a far field spectrum of the array having the        selected parameters; and    -   determining the phase center of the array that minimizes angular        phase variation of the far field spectrum in a predetermined        frequency range.

The present invention also provides a method for determining a phasecenter of a seismic source array, the method comprising:

calculating a far field spectrum of the array at predetermined sphericalangles, and minimizing the phase difference between the farfield spectrawithin a predetermined frequency range by adjusting a vertical referenceposition from which the spherical angles are defined.

Preferably, calculating the far field spectrum comprises calculating theaperture response function, calculating the ghost response function andcalculating the far field spectrum based on a combination of theaperture response function and the ghost response function. Preferably,the angular phase variation is minimized within the range of take-offangles of 0 to 40 degrees. Preferably, the predetermined frequency rangeis 0 to 150 Hz.

Preferably, the array comprises a plurality of layers, each layercomprising a plurality of source elements having substantially the samedepth, and wherein the elements in each layer are configured to firewith a synchronization time delay to align the vertically downgoingwavefields. Preferably, the array comprises two layers of sourceelements and the depth ratio of the two layers is in the range 0.25 to0.6, more preferably in the range 0.3 to 0.5.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described withreference to the accompanying figures, in which:

FIG. 1 illustrates a ghost response of a single depth source, anaperture smoothing function of a 6 element array, combined spectrum forthe 6 element array and spectral difference between the combinedresponse at 22.5 degree and 0 degree take-off angles for a 6 elementlinear array;

FIG. 2 shows the maximum difference of the combined response between22.5 degree take-off angle and 0 degree take-off angle as a function ofthe array length-to-depth ratio of a single depth linear array with N=2,3, 4, . . . elements;

FIG. 3 shows an azimuth-invariant source array;

FIG. 4 compares the isotropic properties of a single depth source with adual depth source;

FIG. 5 shows the difference (in dB) of the dual depth ghost responsebetween the vertical take-off direction and 22.5 degree take-off angleas a function of the depth ratio of the dual sources;

FIG. 6 shows the maximum difference (in dB) between dual depth arrayspectra (combined ghost response and aperture smoothing function) at22.5 deg take-off angle and at normal incidence (0 deg) as a function ofthe array width-to-depth ratio of an array with a linear aperture with 3elements (left panel) and 6 elements (right panel);

FIG. 7 shows the maximum phase difference (in degrees) for the same caseas in FIG. 6;

FIG. 8 shows the maximum difference (in dB) between dual depth arrayspectra (combined ghost response and aperture smoothing function) at 40deg take-off angle and at normal incidence (0 deg) as a function of thearray width-to-depth ratio of an array with a linear aperture with 3elements (left panel) and 6 elements (right panel);

FIG. 9 shows the difference between dual depth ghost response phasespectra at 0 degrees and 20 degrees when the reference point is movedfrom the sea surface to the deepest layer;

FIG. 10 shows the maximum phase difference between the dual depth ghostresponse in the vertical direction and 22.5 degree take-off angle as afunction of the location of the phase center;

FIG. 11 is a flow chart illustrating a method of selecting optimumparameters of a single depth array; and

FIG. 12 is a flow chart illustrating a method of selecting optimumparameters of a multi depth array.

DETAILED DESCRIPTION

The ensuing description provides preferred exemplary embodiment(s) only,and is not intended to limit the scope, applicability or configurationof the invention. Rather, the ensuing description of the preferredexemplary embodiment(s) will provide those skilled in the art with anenabling description for implementing a preferred exemplary embodimentof the invention. It being understood that various changes may be madein the function and arrangement of elements without departing from thespirit and scope of the invention as set forth in the appended claims.

Specific details are given in the following description to provide athorough understanding of the embodiments. However, it will beunderstood by one of ordinary skill in the art that the embodimentsmaybe practiced without these specific details.

As earlier noted, it is desirable that the source array should behave asclosely as possible to a monopole source. Today's commercially availablesources try to achieve this by reducing the array size such that themaximum array dimension is considerable smaller than the shortestwavelength of interest. However, this will not result in a monopolesource spectrum when the sea surface reflection is taken into account.The resulting source will be a dipole. Herein we describe how to designa marine source array that is the optimum approximation to a monopolesource. This is achieved by analysing the relative contribution of thethree main components of the farfield source spectrum: the ghostresponse, the aperture smoothing function, and the monopole spectra ofthe individual source elements. In particular, we observe that only theghost response and the aperture smoothing function is angular dependent,and furthermore, that the power spectral variation with take-off angle(to the vertical) is opposite for the aperture function and the ghostresponse for part of the frequency band, e.g. the aperture functionattenuates non-vertical signals, while the ghost response amplifiessignals propagating at non-vertical angles relative to the verticaloutput.

The general case is presented first. The farfield spectrum of generalmulti-depth source array geometries may be given by

$\begin{matrix}{W = {\sum\limits_{n = 1}^{N}\begin{bmatrix}{{S_{n}(\omega)} \cdot {\exp ( {{j\; k_{x}x_{n}} + {j\; k_{y}y_{n}}} )} \cdot \lbrack {{\exp ( {j\; k_{z}z_{n}} )} + {{\rho exp}( {{- j}\; k_{z}z_{n}} )}} \rbrack} \\{\cdot {\exp ( {- {j\omega\tau}_{n}} )} \cdot {\exp ( {{- j}\; k_{z}z_{ref}} )}}\end{bmatrix}}} & (1)\end{matrix}$

Where S(ω) is the monopole response, x_(n),y_(n),z_(n) are thecoordinates of the N source elements, k_(x),k_(y),k_(z) the respectivespatial wavenumbers, ρ the sea-surface reflection coefficient, ω theangular frequency, and τ_(n) the synchronization delays, e.g. settingτ_(n)=(z_(n)−z_(min))/c aligns the vertically down-going wavefield,where c is the acoustic velocity. Furthermore, the last factor shiftsthe reference point from the origin at the sea surface to a chosenvertical reference position z_(ref). The choice of z_(ref) is importantfor locating the acoustic centre of the array, which will be discussedin detail later.

Note, equation 1 comprises three main factors: the notional sourcespectrum, S(ω), the two-dimensional discrete aperture smoothingfunction, W(k_(x),k_(y)), and the ghost response, W(k_(z)). The twolatter factors may be given by

W _(aperture)(k _(x) ,k _(y))=exp(jk _(x) x _(n) +jk _(y) y _(n))

W _(ghost)(k _(z))=exp(jk _(z) z _(n))+ρ exp(−jk _(z) z _(n))  (2)

In an embodiment of the present invention, by analyzing the relativecontribution of these three factors, improved dimensions may bedetermined for designing isotropic source configurations, i.e. sourcearrays with minimum variation in azimuth- and take-off angle for aspecified frequency band.

Examples of such analyses will be given herein. These examplesdemonstrate that the current practices do not result in the bestisotropic sources. To minimize the source array directivity, today'scommercially available sources aim to approximate a point aperture (zerolateral extent), or at least reduce the array size such that the maximumarray dimension is considerable smaller than the shortest wavelength ofinterest. However, this does not result in optimum isotropic sourceconfigurations when the sea surface reflection is taken into account.

Single Depth Linear Arrays

First, a point aperture is investigated. The farfield spectrum of asingle element at depth d at take-off angle φ is given by

$\begin{matrix}\begin{matrix}{{W( {\omega,\varphi} )} = {{S(\omega)} \cdot \lbrack {{\exp ( {\frac{{j\omega}\; d}{c}\cos \; \varphi} )} - {\exp ( {{- \frac{{j\omega}\; d}{c}}\cos \; \varphi} )}} \rbrack}} \\{= {{j2}\; {{S(\omega)} \cdot {\sin ( {\frac{\omega \; d}{c}\cos \; \varphi} )}}}}\end{matrix} & (3)\end{matrix}$

when the sea-surface reflection coefficient is −1. In this case thefarfield spectrum is simply the product of the notional source spectrum,S(ω), and the ghost response. Note, the farfield spectrum of a pointaperture is not isotropic when the sea-surface reflection is included.It is well-known that its spectrum will have notches at (linear)frequencies that are multiples of f=c/(2 d cos φ).

The spectral variation with take-off angle for a single element may beknown in the prior art, but the prior art does not teach the relation tothe optimum discrete aperture smoothing function.

The spectral variation with take-off angle may be minimized by placingthe N elements at positions such that the associated discrete aperturesmoothing function partially offsets the angular variation of the ghostresponse. In the next example, this is demonstrated for a linear arraywith N elements, symmetric about x=0, for which the farfield spectrum isgiven by

$\begin{matrix}\begin{matrix}{{W( {\omega,\varphi} )} = {j\; 2{{\sin ( {\frac{\omega \; d}{c}\cos \; \varphi} )} \cdot {\sum\limits_{n = 1}^{N}{{S_{n}(\omega)} \cdot {\exp ( {\frac{{j\omega}\; x_{n}}{c}\sin \; \varphi} )}}}}}} \\{= {j\; 2{{\sin ( {\frac{\omega \; d}{c}\cos \; \varphi} )} \cdot {\sum\limits_{n = 1}^{N/2}{2{{S_{n}(\omega)} \cdot {\cos ( {\frac{{j\omega}\; {x_{n}}}{c}\sin \; \varphi} )}}}}}}}\end{matrix} & (4)\end{matrix}$

In a special case of equation 4 where the notional sources, S_(n)(ω),are identical, the farfield spectrum of the linear single depth array isthe product of the ghost response, the discrete aperture smoothingfunction, and the notional source spectrum. Note, since the notionalsource spectrum is not a function of the take-off angle, the notionalsource spectrum is a spectral weighting function for the combinedresponse of the other two factors. Consequently, in certain aspects, forarrays with substantially identical notional source elements, one mayderive the optimum isotropic source array configuration by analysing thecombined response of the ghost response and the aperture smoothingfunction.

FIG. 1 shows an example of this for a linear array with six equidistantelements deployed at the same depth. The top left panel shows the ghostresponse of a single depth source at φ=0° (dash) and φ=22.5° (solid)take-off angles. Note that at the upper half of the frequency band(0.25<d/λ<0.5) the ghost response amplify signals propagating atnon-vertical angles relative to the output at vertical, i.e. the solidcurve is above the dashed. The bottom left panel shows the discreteaperture smoothing function for a 6 element linear equidistant arraywhen the array length is 1.5 times the array depth (top curve), 2.4times the array depth (middle curve), and 3.0 times the array depth(bottom curve). All three curves show the discrete aperture smoothingfunction at φ=22.5°. At normal incidence φ=0° the discrete aperturesmoothing function is constant at 0 dB for all frequencies (not shown).Note that output at non-vertical angles is attenuated relative to theoutput at vertical. Furthermore, note that the attenuation increaseswith frequency and with the aperture length. The top right panel showsthe combined spectrum of the ghost response and the discrete aperturesmoothing function for the 6 element linear array when the array lengthis zero, i.e. point aperture, in the top solid curve, 1.5 times thearray depth (second solid curve), 2.4 times the array depth (third solidcurve), and 3.0 times the array depth (bottom solid curve). All foursolid curves show the combined spectrum at φ=22.5°. Also shown is thecombined spectrum for φ=0° (dashed curve). The bottom right panel showsthe spectral difference in dB between the combined response at φ=22.5°and φ=0° for the 6 element linear array when the array length is zeroi.e. point aperture (top curve), 1.5 times the array depth (secondcurve), 2.4 times the array depth (third curve), and 3.0 times the arraydepth (bottom curve), i.e. the differences between the solid curves anddashed curve in the top right panel. The vertical lines in the top rightand bottom right panels indicate the chosen highest frequency ofinterest (d/λ=0.46). Note that the third from top curve exhibits thesmallest power spectral difference within this bandwidth.

For arrays with elements at substantially the same depth, the sourcedepth, d, is normally chosen such that the highest frequency of interestis smaller than the first notch frequency, i.e. f_(max)<c/(2 d). In thisexample f_(max) is chosen to coincide with the −6 dB point of thevertical ghost response, i.e. wavelengths larger than about 2.17 timesthe depth (d/λ<0.46). As illustrated in the bottom right panel of FIG.1, the attenuation by the discrete aperture smoothing function increaseswith increasing array length and with increasing frequency.Consequently, by configuring the source elements in a linear non-zeroaperture, the angular variation of the aperture smoothing functionpartially offsets that of the ghost response. In this specific examplethe maximum power spectral difference is minimized when the array lengthequals 2.4 times the array depth (third from top curve in bottom rightpanel).

The optimum array length-to-depth ratio also depends on the number ofelements in the linear aperture. This is illustrated in FIG. 2, whichshows the maximum difference (in dB) between the combined spectra at thesame take-off angles (φ=0° and φ=22.5° and within the same bandwidth (−6dB points of the vertical ghost response) as a function of the arraylength-to-depth ratio of a single depth linear array with N=2, 3, 4, . .. elements. As shown in FIG. 2, the optimum length-to-depth ratio is 1.6for a 2-element aperture, 2.0 for a 3-element aperture, and 2.8 for acontinuous aperture. Note, these optimum numbers will be different forother take-off angles and/or other bandwidths. Thus, it can be seen thatthe optimum length to depth ratio is between approximately 1.5 and 3,depending on the number of elements in the array.

Single Depth Planar Arrays

So far only examples of minimizing the spectral variation with take-offangle have been discussed. In an embodiment of the present invention,the variation with azimuth angle may also be minimized when designingoptimum isotropic sources. This may be achieved by applying rotationalsymmetry (in azimuth) to the optimum linear arrays discussed in theprevious section.

An example of such an array, which has six-fold azimuthal symmetry, isdiscussed in J. Hopperstad, J. Synnevaag, and P. Vermeer, 2001, “Anazimuth-invariant source array”: 71st Annual International Meeting, SEG,Expanded Abstract, 52-55, incorporated herein by reference in itsentirety for all purposes. This array configuration is reproduced inFIG. 3 and consists of a circle with two-gun clusters and a three-guncluster in the centre. The volumes are printed next to the airgunpositions. In accordance with an embodiment on the present invention, byapplying the principles discussed herein for single depth linear arrays,the optimum depth for this array may be derived with respect tominimizing the spectral variation with take-off angle: This array has alinear aperture of 3 elements with a total length of 12 metres. Assumingthe same take-off angles and bandwidth of interest as shown in FIG. 2,the optimum length-to-depth ratio for this array is 2.0, i.e. theoptimum depth is 6 metres.

Vertical Line Arrays

For vertical line arrays, equation 1 may be reduced to

$\begin{matrix}{{W( {\omega,\varphi} )} = {\sum\limits_{n = 1}^{N}{{S_{n}(\omega)} \cdot \lbrack {{\exp ( {\frac{{j\omega}\; d_{n}}{c}\cos \; \varphi} )} - {{\rho exp}( {{- \frac{{j\omega}\; d_{n}}{c}}\cos \; \varphi} )}} \rbrack \cdot {\exp ( {- {j\omega\tau}_{n}} )} \cdot {\exp ( {\frac{{- {j\omega}}\; z_{ref}}{c}\cos \; \varphi} )}}}} & (5)\end{matrix}$

As with the single depth source discussed earlier, the best isotropicvertical line array may be derived by investigating the associated ghostresponse when the notional sources are substantially identical. In thiscase, the notional source spectrum is merely a spectral weightingfunction to the multi-depth ghost response.

The next example, FIG. 4, compares the isotropic properties of a singledepth source with a dual depth source. The upper panel of FIG. 4 showsthe ghost response of a point aperture (dashed curves) and a dual depthsource (solid curves) at φ=0° and φ=22.5°. The second panel shows thedifferences (in dB) between the spectra at 0° and 22.5° for the singledepth source (dashed curve) and the dual depth source (solid curve). Thethird panel shows the phase spectra for the curves of the first panel.The chosen vertical reference position is at the sea surface for boththe single depth source and the dual depth source. Staggered firing hasbeen applied to the dual depth source to align the vertically down-goingwavefields. The fourth panel shows the differences between phase spectra(in degrees at 0 degree and 22.5 degree take-off angles. The verticalline in all four panels indicates the chosen highest frequency ofinterest (d_(max)/λ=0.46).

As illustrated in FIG. 4, the variation in spectral power with take-offangle can be greatly reduced by filling the notch associated with thedeepest element. In other words, by selecting the source depths, suchthat the first ghost notch occurs at a frequency significantly higherthan the chosen frequency band, the power spectral difference withtake-off angle is reduced by avoiding the steep roll-off near the notchfrequency. As shown in the second panel in FIG. 4, the dual depth sourcehas extended the upper limit of the frequency band, wherein the powerspectral difference is small, from about d_(max)/λ<0.80 to aboutd_(max)/λ<0.35. Furthermore, by optimizing the depth selection, themulti-depth source can reduce the power spectral difference down to thelevel associated with the zero-frequency notch. This minimum level isgiven by

$\begin{matrix}{{\lim\limits_{\omega->0}\; {\log \frac{{W( {\omega,\varphi} )}}{{W( {\omega,{\varphi = 0}} )}}}} = {\log {{\cos \; \varphi}}}} & (6)\end{matrix}$

FIG. 5 shows the difference (in dB) of the dual depth ghost responsebetween the vertical take-off direction and 22.5 degree take-off angleas a function of the depth ratio of the dual sources. FIG. 5 shows thatin certain aspects this minimum level may be achieved for the dual depthsource when the depth ratio is between 0.3 and 0.5 (at 22.5 deg take-offangle when the bandwidth of interest is given by the −6 dB points of thevertical ghost response of the deepest element). The minimum level isgiven by 20·log 10(cos(22.5°))=0.69 dB.

Multi-Depth Arrays

In an earlier example, it was demonstrated that single depth planararrays need to tailor the aperture smoothing function to the ghostresponse in order to minimize the spectral variation with take-offangle. However, single depth arrays are sensitive to geometryperturbations, such as changes to the length-to-depth ratio of thearray.

With multi-depth arrays the optimized array configuration can be madepractically insensitive to the horizontal array dimension. This may beachieved by combining the source depths such that the first notch in themulti-depth ghost response is far beyond the highest frequency ofinterest. FIGS. 6 to 8 show examples of for what maximum horizontaldimension of the arrays, because of the associated aperture smoothingfunction, starts affecting the maximum difference of the farfieldspectra. In all examples, the bandwidth of interest is given by the −6dB points of the vertical ghost response of the deepest element.Furthermore, in all examples the depth ratio, d_(min)/d_(max), is 0.5.FIG. 6 shows the power spectral difference between vertical and 22.5degrees take-off angle, and FIG. 8 shows the difference between verticaland 40 degrees take-off angle. The left panel of FIG. 6 and FIG. 8 showsthe case with three elements in the linear aperture, while the rightpanels show the six element case.

The darkest contours (<2 dB) in FIG. 6 and FIG. 8 show the levelcorresponding to the power spectral difference associated with thezero-frequency notch. This absolute minimum level is achieved when thelength-to-depth ratio is less than about two, i.e. when the total arrayconfiguration fits within a ±45° cone from the sea surface.

In one embodiment of the present invention, by combining the concept ofthe azimuth invariant source array as provided in J. Hopperstad, J.Synnevaag, and P. Vermeer, 2001, “An azimuth-invariant source array”:71st Annual International Meeting, SEG, Expanded Abstract, 52-55 withthe take-off angle invariant configurations discussed herein, the bestisotropic source arrays may be configured, i.e. arrays with minimumvariation in both azimuth and take-off angle; e.g. deploying two of thesource array shown in FIG. 3, so that the depth ratio of the two layersis between 0.3 and 0.5, and so that the maximum lateraldimension-to-depth ratio is less than two. The resulting farfieldspectrum will have optimum isotropy for take-off angles up to 22.5° andfrequencies below the −6 dB points of the vertical ghost response of thedeepest element.

The present invention provides a method for determining optimumparameters for source arrays that offsets the angular variation of theghost response with the aperture smoothing function and the notionalsource spectra. It has been shown that the isotropy of single depthplanar arrays is limited by how well the lateral geometry can betailored to the angular variation of the high-frequency ghost notch,while multi-depth arrays can be designed to be insensitive to thelateral dimension. Optimum isotropic multi-depth sources are limited bythe difference in power spectra in the region of the zero-frequencynotch rather than the high frequency notch. This results in a smallervariation with take-off angle than of the single depth arrays.

Finding the Acoustic Centre of a Multi-Depth Array

The above analysis relates to achieving improved isotropy of the powerspectrum of the source array. However, the phase spectrum of the sourcearray also varies with azimuth and take-off angle. In the followingexamples we demonstrate that the phase variation with take-off anglestrongly depends on the point that is considered the origin of thewavefield. This origin is often referred to as the acoustic center ofthe array. The acoustic center is the point from which the signalappears to have radiated. The center might be different for differentfrequencies; the one that minimizes the phase error across the wholebandwidth is termed the “phase center”. The phase center of a singledepth array is at the sea surface, because of the symmetry of the directarrival of the “real array” and the sea surface reflection of the “ghostarray”. For multi-depth arrays, the optimum location of the center iscomplicated, because of the firing time delay that is applied to thedeeper elements in order to align the vertically propagating wavefields.The staggered firing introduces significant phase variations withtake-off angle. This is illustrated in the bottom panel of FIG. 4, whichshows that aligning the vertical down-going wavefield results in a phaseshift of up to ten degrees (between 22.5° take-off angle and thevertical direction). In the examples discussed in this document, it isfound that the optimum location (i.e., the phase center) varies from thedepth of deepest source layer to about the same depth as the shallowestlayer, depending on the ratio of the depths.

The farfield spectrum of general multi-depth geometries is given byequation (1) above, wherein τ_(n) is the synchronization delays; e.g.,setting τ_(n)=(z_(n)−z_(min))/c aligns the vertically downgoingwavefield, where c is the acoustic velocity. Note, the last factorshifts the reference point from the origin at the sea surface to achosen vertical reference position z_(ref), from which the traveltimesare computed. The optimum choice of reference position is termed the“phase center”. We discuss the optimum choice for z_(ref), i.e., thevertical position of phase center, such that the phase variation withtake-off angle is minimized.

In the special case of identical planar source arrays that arevertically distributed, and where the triggering of the elements arestaggered to align the vertically propagating wavefields, e.g., twoidentical layers deployed in an over/under configuration, Equation 1 canbe expressed as:

$\begin{matrix}{W = {\sum\limits_{i = 1}^{L}{{\begin{pmatrix}{( {{\exp ( {\frac{{j\omega}\; d_{I}}{c}\cos \; \varphi} )} + {{\rho exp}( {\frac{{- {j\omega}}\; d_{I}}{c}\cos \; \varphi} )}} ) \cdot} \\{\exp ( {{- {j\omega}}\frac{d_{I} - d_{\min}}{c}} )}\end{pmatrix} \cdot \exp}{( {\frac{{- {j\omega}}\; z_{ref}}{c}\cos \; \varphi} ) \cdot {\sum\limits_{m = 1}^{M}{{S_{m}(\omega)} \cdot {\exp ( {{j\; k_{x}x_{m}} + {j\; k_{y}y_{m}}} )}}}}}}} & (7)\end{matrix}$

where d is the depth of each layer, φ is the take-off angle, and M isthe number of elements in each of the L depth layers. Equation (7) showsthat the farfield spectrum is, in this case, simply the product of themulti-depth ghost response (including the phase delay for the staggeredtriggering), the shift factor for the chosen vertical reference positionz_(ref), and the farfield spectrum of the identical depth layers.Similarly, note that the farfield phase spectrum of the array is the sumof the individual phase spectra of these three factors. Consequently,source arrays with isotropic phase spectra can be synthesized byanalyzing the relative contribution of the phase spectra of themulti-depth ghost response, the vertical shift factor, and the identicaldepth layers. In particular, the following section discusses the optimumchoice for z_(ref), i.e., the vertical position of phase center.

Above it is demonstrated that the angular variation of the source powerspectrum can be significantly reduced by using a second depth layer. Itcan also be showed that there is no advantage in using more than twolayers (for reducing the angular variation). An example of the dualdepth improvement is shown in the top two panels of FIG. 4, whichcompare the ghost response of a single depth and that of a dual depthsource. The power spectrum of this dual depth source has optimumisotropy, because the take-off angle variation is limited by thedifference near the zero-frequency notch. In contrast, the powerspectrum of the single depth source is limited by the difference in theregion close to the high-frequency notch.

Symmetric single depth arrays are better than any multi-depth deptharrays in terms of isotropy of the phase spectrum (but not the powerspectrum or the pulse shape). The bottom panel of FIG. 4 shows anexample of this, where the vertical reference, z_(ref), is located atthe sea surface. The difference between the single depth phase spectra(dashed lines) at 22.5° and 0° take-off angle is zero up to thefrequency of the first ghost notch. While for the corresponding dualdepth case, the staggered triggering results in a phase difference of upto 10° within the specified bandwidth.

The phase difference with take-off angle can be reduced by optimizingthe location of the chosen vertical reference. FIG. 9 shows how thephase difference of the bottom panel of FIG. 4 (d_(min)/d_(max)=0.5) isreduced by relocating the vertical reference from the sea surface(lowest curve) towards the largest source depth (top curve). The maximumphase difference, within the specified bandwidth, is minimized whenz_(ref)≈0.55 ·d_(max), (second curve from top), i.e., the phase centeris located slightly deeper than the shallowest source depth. In thisexample, the maximum phase error is reduced from 10° to 3.2°.Alternatively, one can, in theory, completely correct for this phasedifference in data processing by using a different phase center for eachfrequency. In these examples we only discuss using one phase center forthe entire bandwidth.

In the analysis above, it is shown that, within this bandwidth andbetween these take-off angles, the isotropy of the dual depth sourcepower spectrum is optimized when the depth ratio, d_(min)/d_(max), isbetween 0.3 and 0.5.

The optimum location of the vertical reference, i.e., the phase center,within this range of depth ratios is illustrated in FIG. 10 (between22.5° take-off angle and vertical when the bandwidth of interest isgiven by the −6 dB points of the vertical ghost response of the deepestelement). For a depth ratio of 0.3, the phase difference between 22.5°take-off angle and vertical is minimized when the reference point islocated slightly deeper than the largest source depth (z_(ref)=1.05d_(max)). For a depth ratio of 0.4, z_(ref)=0.75 d_(max). For theexample in FIG. 4 and FIG. 9 (d_(min)/d_(max)=0.5), the maximum phasedifference is reduced from 10.0 to 3.2 degrees when the reference pointis shifted from the sea surface to 0.55 times the depth of the deepestelement.

Furthermore, FIG. 10 shows that the phase center, i.e., the optimumz_(ref), moves towards the sea surface (z_(ref)/d_(max)=0) as thevertical distance between the two source layers decreases. A smallerlinear phase shift is required to compensate for the angular variationcaused by the vertical alignment of the wavefields (the staggeredtriggering). The phase center of a single depth array is at the seasurface, as previously discussed.

The extent of these optimized results are illustrated in FIGS. 6 and 7,which show the effect of the array length in an over/under sourceconfiguration, where each layer comprises a three or six element linearaperture. In FIG. 7, the phase centre has been shifted to its optimumlocation for these take-off angles and this bandwidth, i.e. 0.55 timesthe depth of the deepest element. FIG. 6 shows the difference betweenthe power spectra, and FIG. 7 shows the difference between the phasespectra. Above, it is shown that the effect of the lateral geometry, onthe power spectra, is insignificant when the length is less than abouttwo times the depth. This region also coincides with the small contoursof the phase difference, although the phase difference is minimized whenthe array lengths are equal. Note, the maximum phase variation isdominated by the staggered firing, rather than the different aperturesmoothing functions, when the array length-to-depth ratio is less thanabout 1 (the 4° phase contour).

The variation with azimuth angle must also be minimized when designingoptimum isotropic sources. This can be achieved by applying rotationalsymmetry (in azimuth) to the optimum linear arrays discussed in theprevious example. An example of such a configuration, which has six-foldazimuthal symmetry, is the source array by Hopperstad et al. (2001)where two such hexagonal arrays are deployed in an over/underconfiguration, where one such depth layer is shown in FIG. 3. Theresulting source will have a directivity pattern that is invariant inazimuth and minimized variation in take-off angle.

We have shown that the acoustic center of a single depth array is at thesea surface when the surface reflection is considered to be part of thesource signature. This is the position that minimizes the phasedifference with take-off angle, hence the term “phase center”.Furthermore, we have shown that by careful choice of array elementpositions and firing delays, a source can be designed whose signaturevaries very little with take-off angle or azimuth. However, thismulti-depth isotropic source has a phase center that is not located atthe surface. In the examples discussed here, the phase center is locatedsomewhere between the shallowest and deepest elements at a position thatdepends on the details of the array and the desired bandwidth. Theposition of the phase center must be included in the processing of theseismic image.

FIG. 11 shows an embodiment of the process of selecting the optimumparameters for a single depth array, such that the spectral variation isminimized between the maximum take-off angle of interest, φ_(max), andvertical (φ=0) within the selected frequency band.

In step 1 of FIG. 11, in step 1 the frequency band of interest isselected. In step 2, the source depth is selected based on the maximumfrequency of interest, e.g. such that the maximum frequency coincideswith the −6 dB point of the vertical ghost response. In step 3, therange of take-off angels, φ, of interest and the range of azimuthangles, θ, of interest, e.g. 0≦φ≦22.5° and 0≦θ<360.

In step 4, the aperture function is calculated by selecting the numberof elements in the array and the lateral (x,y) position of each element.The element positions should have a degree of rotational symmetry withinthe azimuth angles of interest.

In step 5, a set of notional sources is selected and assigned to theaperture positions. The notional sources can be chosen based onconventional source design criteria, such as minimum required output andspectral flatness of the vertical farfield spectrum. In addition, thenotional sources should be assigned to element positions, so thatsubstantially identical notional sources are symmetrically locatedaround the center.

In step 6, the difference of the power spectrum given by equation 1 iscalculated, ie. the maximum dB difference between the maximum andminimum take off angles, eg φ=22.5 and vertical (φ=0) at all azimuthangles.

In step 7, the result of step 6 is compared with a user-definedtolerance, e.g. that the maximum power spectral difference for a givenazimuth angle is minimized, i.e. maximum difference symmetricallydistributed within the frequency band of interest within the numericalaccuracy.

If the result is not less than the user defined tolerance, then in step8 the lateral dimensions are adjusted by adjusting the x,y-positions ofthe elements in the aperture function, e.g. scale the aperture by aconstant factor, and then steps 5, 6 and 7 are repeated until the resultof step 7 is within the tolerance, wherein the optimum dimensions havebeen found.

FIG. 12 shows an embodiment of the process of selecting the optimumparameters for a multi-depth array, such that the spectral variation isminimized between the maximum take-off angle of interest, φ_(max), andvertical (φ=0) within the selected frequency band.

In step 1, the frequency band of interest is selected. In step 2, thesource depths, and in particular the ratio of depths, are selected, suchthat the first notch frequency in the multi-depth ghost response islocated outside the frequency band of interest, i.e., such thatf_(max)<c/(2·d_(min)), e.g. so that f_(max)≈c/(2·d_(max)). In step 3,the range of take-off angels, φ, of interest and the range of azimuthangles, θ, of interest, e.g. 0°≦φ≦22.5° and 0°≦θ<360° are selected.

In step 4, the difference of the power spectrum of the multi-depth ghostresponse is calculated, between the maximum take off angle of interestand vertical, e.g. the maximum dB difference between φ=22.5° andvertical φ=0° at all azimuth angles. In step 5, this difference iscompared with a user-defined tolerance. If the difference is greaterthan the tolerance, at step 6 the source depths are adjusted and step 4is repeated until a difference within the tolerance is achieved e.g.that the maximum power spectral difference for a given azimuth angle isminimized, i.e. maximum difference symmetrically distributed within thefrequency band of interest within the numerical accuracy.

Once the optimum source depths have been selected, in step 7 theaperture function for each depth layer is selected, by selecting thenumber of elements at each depth layer and the lateral (x,y) position ofeach element. The element positions should have a degree of rotationalsymmetry within the azimuth angles of interest.

At step 8, the difference in power of the combined spectrum of themulti-depth ghost response and the aperture function between the maximumtake-off angle off interest (e.g.) φ=22.5° and vertical (φ=0° at allazimuth angles is calculated, and at step 9 this is compared with auser-defined tolerance. If the difference is greater than the tolerance,the lateral dimensions of the array are adjusted at step 10 (e.g. byscaling the aperture by a constant factor), and step 8 is repeated untilthe difference is less than the tolerance. This means that the maximumpower spectral difference for a given azimuth angle is minimized, i.e.maximum difference symmetrically distributed within the frequency bandof interest within the numerical accuracy.

At step 11, a set of notional sources is selected and assigned to theaperture positions of each depth layer. The notional sources can bechosen based on conventional source design criteria, such as minimumrequired output and spectral flatness of the vertical farfield spectrum.In addition, the notional sources should be assigned to elementpositions, so that substantially identical notional sources aresymmetrically distributed at each depth layer.

At step 12, the difference in phase of the total spectrum of the sourceis calculated, as described in equation 1, between the maximum take-offangle of interest (eg φ=22.5°) and vertical (φ=0°) at all azimuthangles. In the initial calculation use the sea-surface as the verticalreference, i.e. z_(ref)=0. At step 13, this is compared withuser-defined tolerance, and if the difference is greater than thetolerance, z_(ref) is increased and step 12 repeated until a differencewithin the tolerance is achieved. Thus the maximum phase difference fora given azimuth angle is minimized, i.e. maximum differencesymmetrically distributed within the frequency band of interest withinthe numerical accuracy.

At step 15, the optimum source parameters have been found, i.e. theoptimum depths, optimum lateral positions, and optimum verticalreference position (i.e. the phase center).

While the principles of the disclosure have been described above inconnection with specific apparatuses and methods, it is to be clearlyunderstood that this description is made only by way of example and notas limitation on the scope of the invention. Further, a number ofvariations and modifications of the disclosed embodiments may also beused.

1. A method for selecting parameters of a seismic source arraycomprising a plurality of source elements each having a notional sourcespectrum, the method comprising: calculating a ghost response functionof the array; calculating directivity effects of the array; andadjusting the parameters of the array such that the directivity effectsof the array are compensated by the ghost response to minimize angularvariation of a far field response in a predetermined frequency range. 2.The method of claim 1, wherein the depths of the source elements areselected such that a first notch in the ghost response is above thepredetermined frequency band.
 3. The method of claim 1, wherein thearray comprises a single layer of source elements at a common depth, andthe parameters comprise the length to depth ration of the source array.4. The method of claim 3, wherein the length to depth ratio is in therange 1.5 to
 3. 5. The method of claim 1, wherein the array comprises aplurality of layers, each layer comprising a plurality of sourceelements having substantially the same depth.
 6. The method of claim 5,wherein the minimum wavelength of the predetermined frequency band isgreater than 4/3 of the maximum source element depth, i.e. d_(max)/λ,0.75
 7. The method of claim 5, wherein the minimum wavelength of thepredetermined frequency band is greater than two times the maximumsource element depth, i.e. d_(max)/λ, 0.5
 8. The method of claim 5,wherein the depth ratio of the layers is adjusted such that a firstnotch in the ghost response is above the predetermined frequency band.9. The method of claim 8, wherein the array comprises two layers ofsource elements and the depth ratio of the two layers is in the range0.25 to 0.6.
 10. The method of claim 9, wherein the array comprises twolayers and the depth ratio of the two layers is in the range 0.3 to 0.5.11. A method according to claim 5, where the notional source spectra ofthe elements at each depth layer are substantially identical.
 12. Themethod of claim 5, wherein the length to depth ratio of each layer isless than
 2. 13. The method of claim 1, where the element positionssubstantially form a vertical line array.
 14. A method of claim 1, whereall of the notional source spectra are substantially identical.
 15. Amethod of claim 1, wherein the positions of the source elements haverotational symmetry in azimuth of order three or greater in thehorizontal plane.
 16. The method of claim 1, wherein the angularvariation is minimized within the range of take-off angles of 0 to 40degrees.
 17. The method of claim 1, wherein the predetermined frequencyrange is 0 to 150 Hz.
 18. The method of claim 5, further comprising:calculating a far field spectrum of the array having the selectedparameters; and determining the phase centre of the array that minimizesangular phase variation of the far field spectrum in a predeterminedfrequency range.
 19. A method for determining a phase center of aseismic source array, the method comprising: calculating a far fieldspectrum of the array at predetermined spherical angles, and minimizingthe phase difference between the farfield spectra within a predeterminedfrequency range by adjusting a vertical reference position from whichthe spherical angles are defined.
 20. The method of claim 19, whereincalculating the far field spectrum comprises calculating the apertureresponse function, calculating the ghost response function andcalculating the far field spectrum based on a combination of theaperture response function and the ghost response function.
 21. Themethod of claim 19, wherein the angular phase variation is minimizedwithin the range of take-off angles of 0 to 40 degrees.
 22. The methodof claim 19, wherein the predetermined frequency range is 0 to 150 Hz.23. The method of claim 19, wherein the array comprises a plurality oflayers, each layer comprising a plurality of source elements havingsubstantially the same depth, and wherein the elements in each layer areconfigured to fire with a synchronization time delay to align thevertically downgoing wavefields.
 24. The method of claim 23, wherein thearray comprises two layers of source elements and the depth ration ofthe two layers is in the range 0.25 to 0.6.
 25. The method of claim 23,wherein the array comprises two layers and the depth ratio of the twolayers is in the range 0.3 to 0.5.
 26. The method of claim 18, furthercomprising processing seismic data using the determined phase centre ofthe array.
 27. A method for selecting parameters of a seismic sourcearray comprising a plurality of source elements each having a notionalsource spectrum so that the directivity effects of the sea surfacereflection are cancelled by the directivity effects of the arraypositions, firing times and notional source spectrums.